斗鸡 is a separable verb-object compound (离合词) that can function as a noun meaning 'cockfighting (the activity)' or 'gamecock (a fighting rooster)'. As a verb-object structure, it can be split — for example, 斗了一次鸡 ('fought cocks once') — with the verb 斗 taking aspect markers or complements and 鸡 acting as the object. Use it as a noun when referring to the sport or the bird, and as a split verb when describing the action.